- Research Article
- Open Access
Slowly Oscillating Solutions of a Parabolic Inverse Problem: Boundary Value Problems
© F. Yang and C. Zhang. 2010
- Received: 11 October 2010
- Accepted: 20 December 2010
- Published: 29 December 2010
The existence and uniqueness of a slowly oscillating solution to parabolic inverse problems for a type of boundary value problem are established. Stability of the solution is discussed.
- Differential Equation
- Continuous Function
- Integral Equation
- Partial Differential Equation
- Unique Solution
It is well known that the space of almost periodic functions and some of its generalizations have many applications (e.g., [1–13] and references therein). However, little has been done for to inverse problems except for our work in [14–16]. Sarason in  studied the space of slowly oscillating functions. This is a -subalgebra of , the space of bounded, continuous, complex-valued functions on with the supremum norm . Compared with , is a quite large space (see [17–20]). What we are interested in is based on the belief that certainly has a variety of applications in many mathematical areas too. In , we studied slowly oscillating solutions of a parabolic inverse problem for Cauchy problems. In this paper, we devote such solutions for a type of boundary value problem.
Set . Let (resp., , where ) denote the -algebra of bounded continuous complex-valued functions on (resp., ) with the supremum norm. For (resp., ) and , the translate of by is the function (resp., , ).
A function is said to be slowly oscillating in and uniform on compact subsets of if for each and is uniformly continuous on for any compact subset . Denote by the set of all such functions. For convenience, such functions are also called uniformly slowly oscillating functions.
As in , we always assume that is uniformly continuous.
The following two propositions come from [15, Section 1].
The following proposition shows that the composite is also slowly oscillating.
be the fundamental solution of the heat equation .
To show the main results of this section, the following lemmas are needed. The first lemma is Lemma 3.1 on page 15 in .
Apply Lemma 2.1 to get the conclusion.
The existence and uniqueness of the solution comes from Theorem 5.3 on page 320 in .
By Lemma 2.1, one gets the desired inequality.
Consider the following problem.
Problems 1, 2, and 3 are equivalent to each other.
the equivalence of Problems 2 and 3 can be proved similarly. The proof is complete.
One can directly test that Problem 3 is equivalent to (2.37)-(2.38).
The proof is complete.
Under the conditions in Theorem 2.6, the solution of Problem 3 is unique.
The research is supported by the NSF of China (no. 11071048).
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